It is well known that orthodox quantum mechanics does not make unambiguous predictions for the statistics in arrival time (or time-of-flight) experiments. Bohmian mechanics (or de Broglie--Bohm theory) offers a distinct conceptual advantage in this regard, owing to the well-defined concepts of point particles and trajectories embedded in this theory. We revisit a recently proposed experiment [S. Das and D. D\"urr, Sci. Rep. 9, 2242 (2019)], the numerical analysis of which revealed a striking spin dependence in the (Bohmian) time-of-arrival distributions of a spin-1/2 particle. We present here a mathematically tractable variant of the same experiment, where the predicted effects can be established rigorously. We also obtain some results that can be compared with experiment.
We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
We discuss Bohmian paths of the two-level atoms moving in a waveguide through an external resonance-producing field, perpendicular to the waveguide, and localized in a region of finite diameter. The time spent by a particle in a potential region is not well-defined in the standard quantum mechanics, but it is well-defined in the Bohmian mechanics. Bohm's theory is used for calculating the average time spent by a transmitted particle inside the field region and the arrival-time distributions at the edges of the field region. Using the Runge-Kutta method for the integration of the guidance law, some Bohmian trajectories were also calculated. Numerical results are presented for the special case of a Gaussian wave packet.
In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
According to Standard Quantum Mechanics (SQM), known as the Copenhagen Interpretation, the complete description of a system of particles is provided by its wave function. However, in the de Broglie-Bohm theory of Bohmian Quantum Mechanics (BQM), the additional element which is introduced apart from the wave function is the particle position, conceived in the classical sense as pursuing a definite continuous track in space-time. In BQM formulation, depending on the configuration of the potential barrier and the energy of the packet, the particle trajectories have been shown to take distinct paths. We will consider several barrier heights and show that in a Bohmian interpretation of the problem, there is no such thing as Quantum Tunnelling.